Determining the Support of a Random Variable Based on its Characteristic Function

Abstract

In this article, we consider determining the support of a random variable X if only its corresponding characteristic function, ϕ_X (t) = E(e^itX) , is known, where the support is the set of all its possible realizations.i.e. S = {x; f (x) > 0} . In other words, the amount of information contained in characteristic function about the support of a random variable is investigated. The two main components of the probability distribution on any random variable are: S and f (x). Most of the proposed methods in the literature focused on determining only the density (mass) function, f (x), assuming S is known. No one has yet considered retrieving the support, S, from the corresponding characteristic function before estimating f (x). This paper is an attempt to complete the gap by retrieving S before estimating f (x), especially when the underlying random variable is a mixture of both  discrete and absolutely continuous random variables. It is found that the tail behavior of ϕ_X (t) reveals most of the information about S. The theorems relating the properties of ϕ_X (t) to S are utilized to formulate the proposed method. Several examples are listed for illustrating the usefulness of the studied method.

Key words and phrases. Characteristic Function, Distribution theory, Random variable Support.

2000 Mathematics Subject Classification. 45B05, 45B99